Download piCore-5.3.1-boot.tar.gz extract to a temporary folder and overwrite content of /mnt/mmcblk0p1 after mounting it. It will overwrite custom boot options. If you have customized cmdline.txt on config.txt in the old system, do it again. Do not reboot if using wireless remote connection, otherwise you will loose connection due to incompatible kernel modules!

Next download kernel module tcz's from the repo matching new kernel version and reboot. You must have a new, updated system working. It is not needed updating from piCore-5.3

Please note, there may be kernel module extensions which are not yet available in the repo.

Before modifying mmcblk01 content, it is adviced to make backup to a persistent device.